The payoff of a floating lookback call option is indeed the final current price of an asset minus the minimum or lowest asset price that the underlying asset has achieved during the life of the option. This is because a floating lookback call option allows the holder to 'look back' over the life of the option and select the lowest price as the strike price. This means that the holder can benefit from the maximum possible upward movement of the asset's price. This type of option provides a safety net for the holder, as they can choose the most favorable price during the option's life, maximizing their potential profit.

Choice B is incorrect. A floating lookback put option's payoff is calculated as the maximum asset price during the life of the option minus the final current price of the asset, which does not match with our given payoff structure.

Choice C is incorrect. A fixed lookback call option's payoff is calculated as the final current price of the asset minus a predetermined strike price or alternatively, it can be calculated as maximum value of underlying asset during life of option minus strike price. This also does not align with our given payoff structure.

Choice D is incorrect. A fixed lookback put option's payoff would be either zero or strike price minus minimum value of underlying asset during life of option, whichever is higher. This too doesn't correspond to our given exotic options' payoff structure.